For the Spin-orbit Hamiltonian at N\(^3\) LO , we remove both the divergent terms as well as terms involving logarithms by finding a suitable canonical transformation. Inspired by [1]}, we build the following ansatz for the infinitesimal
generator
\(\mathcal {G}_{\rm SO-N^3LO}=\left(\frac{1}{c^9}\right)&\Bigg \lbrace \frac{G_N^3}{r^3}\left[g_3 \frac{1}{m_{(1)}m_{(2)}} \Big (\textbf {S}_{(1)}\cdot (\textbf {p}_{(1)}\times \textbf {p}_{(2)})\Big ) + g_4 \frac{1}{m_{(1)}m_{(2)}} \Big (\textbf {S}_{(2)}\cdot (\textbf {p}_{(1)}\times \textbf {p}_{(2)})\Big )\right]\nonumber \\&+\frac{G_N^3}{r^3}\Bigg [\frac{1}{m_{(1)}} \Big (\textbf {S}_{(1)}\cdot (\textbf {p}_{(1)}\times \textbf {n})\Big ) \left(g_{5} \frac{1}{m_{(1)}} \left(\textbf {p}_{(1)} \cdot \textbf {n}\right) + g_{6} \frac{1}{m_{(2)}} \left(\textbf {p}_{(2)} \cdot \textbf {n}\right)\right)\nonumber \\&\quad \quad \quad +\frac{1}{m_{(2)}} \Big (\textbf {S}_{(1)}\cdot (\textbf {p}_{(2)}\times \textbf {n})\Big ) \left(g_{7} \frac{1}{m_{(1)}} \left(\textbf {p}_{(1)} \cdot \textbf {n}\right) + g_{8} \frac{1}{m_{(2)}} \left(\textbf {p}_{(2)} \cdot \textbf {n}\right)\right)\nonumber \\&\quad \quad \quad +\frac{1}{m_{(1)}} \Big (\textbf {S}_{(2)}\cdot (\textbf {p}_{(1)}\times \textbf {n})\Big ) \left(g_{9} \frac{1}{m_{(1)}} \left(\textbf {p}_{(1)} \cdot \textbf {n}\right) + g_{10} \frac{1}{m_{(2)}} \left(\textbf {p}_{(2)} \cdot \textbf {n}\right)\right)\nonumber \\&\quad \quad \quad +\frac{1}{m_{(2)}} \Big (\textbf {S}_{(2)}\cdot (\textbf {p}_{(2)}\times \textbf {n})\Big ) \left(g_{11} \frac{1}{m_{(1)}} \left(\textbf {p}_{(1)} \cdot \textbf {n}\right) + g_{12} \frac{1}{m_{(2)}} \left(\textbf {p}_{(2)} \cdot \textbf {n}\right)\right) \Bigg ] \Bigg \rbrace \, ,\) 
